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mathematics through inquiry
Additional essays on teaching mathematics through inquiry (C1)
(This essay was adapted from: R. Borasi (1992). Learning mathematics through inquiry. Portsmouth, NH: Heinemann (pp.158-164), for inclusion in the multi-media package "Introducing math teachers to inquiry: Framework and supporting materials to design professional development" [Borasi & Fonzi, 1998])
For many people, mathematics can be reduced to a collection of pre-established facts, rules, and techniques essentially having to do with numbers and (at best) geometric shapes. This is not surprising if we consider the content of current precollege school mathematics curricula. Unfortunately, the following description captures well the kind of mathematics that many students experience as a result of their schooling:
Arithmetic computation is entrenched as the basis of the mathematics curriculum, with the "four" rules gradually being developed to handle more and more complicated "numbers"-natural, integer, fractions, decimal, complex and, later, matrices and vectors. Algebraic work develops the skills of solving more and more complicated "equations" and of rearranging complicated expressions so that they can be "solved". Geometry, if taken seriously at all, is developed as an area to which one can apply arithmetical and algebraic techniques, be it thereby trigonometry or coordinate geometry. And for those who have succeeded at, or survived, that diet, the gateway to further delight is the calculus, with its myriad of integrals and differential equations waiting to be recognized, classified, and of course, "solved". (Bishop, 1988, p. 7)
As a result of their experience of mathematics in school, most people (not just young students) conceive of mathematics as "the discipline of certainty" par excellence, consisting only of indisputable facts and techniques (Resnick, 1988, p. 32). However reasonable such a conception of mathematics may at first seem, many mathematicians and mathematics educators have argued that it does not reflect the real nature of mathematics. I agree with this contention and want to propose an alternative view of mathematics as a humanistic discipline.
The activity of applied mathematicians today challenges the perception of mathematics as a well­structured domain (Pollak, 1970). Whether they are asked to deal with real­life situations, such as improving the public transportation of a certain city, or to propose models that can help us predict complex phenomena, such as the weather, these mathematicians face problems that are very different from those encountered in mathematics textbooks, even those at the college level. These professionals are rarely, if ever, presented with a well­defined problem and expected to apply known mathematical methods to come up with an objective solution. Rather, the task is more often presented to them in the more vague and open­ended form of a "problematic situation"-a situation the client is not satisfied with and would like to improve, such as the perceived inefficiency in a city's public transportation. The first and most crucial step for an applied mathematician is to define more specific problems that can be approached with mathematical tools and whose solution can help achieve the original goal. Obviously, a good understanding of the context-the whole complexity of the problematic situation presented to them as well as the goals and values of the client who proposed it-is essential at this stage. Since the way problems are framed will determine the type of solutions sought for, the evaluation of the solutions obtained cannot be made solely on the basis of the appropriate choice and execution of algorithms. It will have to take into account other factors as well, such as the client's satisfaction with the proposed solution, the relationship between the costs and the benefits of its implementation, and even the acceptability of its potential consequences in light of the mathematician's own values and beliefs.
The complexity and nonlinearity of mathematical applications is a reality that affects not only the work of professional mathematicians but also the activity of most business employees and even the daily lives of people. Recent studies by anthropologists interested in everyday cognition have revealed that people's spontaneous use of mathematics in their daily lives has more in common with the activity of professional mathematicians as I have described it than with the tasks traditionally assigned in school mathematics (Rogoff and Lave, 1984). Consider, for example, the following description of how shoppers use arithmetic in the routine task of selecting grocery items in a supermarket:
Although arithmetic problem­solving plays various roles in grocery shopping, its preponderant use is for price comparison. This kind of calculation occurs at the end of largely qualitative decision­making processes which smoothly reduce numerous possibilities on the shelf to single items in the cart. A snag occurs when the elimination of alternatives comes to a halt before a choice has been made. Arithmetic problem­solving is both an expression of and a medium for dealing with these stalled decision processes. It is, among other things, a move outside the qualitative characteristics of a product to its characterization in terms of a standard of value, money.... Shoppers are not comparing prices merely to gain information that will then be weighted appropriately with respect to other information, such as other features of competing brands. Rather, shoppers explicitly compare prices only when they have no strong preference among brands.... The routine nature of grocery­shopping activity and the location of price arithmetic at the end of decision­making processes suggest that the shopper must already assign rich content and shape to a problem solution by the time arithmetic becomes an obvious next step. Problem solving under these circumstances is an iterative process. It involves, on the one hand, what the shopper knows and what the setting holds that might help and, on the other hand, what the solution looks like. The activity of finding something problematic subsumes a good deal of knowledge about what would constitute a solution. In the course of grocery shopping many of a problem solution's parameters are marshalled into place as part of the process of deciding, up to a point, what to purchase. (Lave, Murtaugh, and de la Roche, 1984, pp. 81-83)
As this discussion makes clear, mathematical applications require not only good technical knowledge but also the ability to take into account the context in which one is operating, the purpose of the activity, the possibility of alternative solutions, and also personal values and opinions that can affect one's decisions. Unfortunately, none of these elements is usually recognized as relevant to mathematical activity by people who have gone through traditional schooling.
I am well aware of the objection that some may want to raise at this point: "But this has nothing to do with mathematics, it has to do only with its applications!" The debate on the distinction between "pure" and "applied" mathematics, and on the legitimacy of the latter to be considered mathematics proper, has indeed a long history in the field (Halmos, 1981; Poston, 1981). Although I do not wish to get into it here, I would like to point out that conceptually separating mathematical knowledge from its uses can have dangerous repercussions when discussing the learning and teaching of mathematics, as recent research on situated learning has suggested (Lave, 1988). For the scope of this discussion, let it suffice to say that, since one of the major goals of school mathematics is to empower students to use mathematics appropriately, schools have the responsibility to make students aware of the ambiguous and contextualized nature of "real­life" mathematical problems.
Regardless of one's conclusions about whether the application of mathematics constitutes proper mathematical activity or not, it is important to realize that the realm of "pure" mathematics, too, is far from achieving the ideals of objectivity, certainty, absolute truth, and rigor usually associated with this discipline. The way mathematics has developed historically challenges these common expectations.
Perhaps because textbooks and lectures tend to present mathematical results in a "neat" and organized way, few people realize that those results have not always been achieved in a straightforward manner. On the contrary, historical accounts like those of Morris Kline (1980, 1985) reveal the centuries of intellectual struggle that were needed to produce even the most fundamental mathematical results, such as the number systems middle school students work with today. The development of certain topics, such as infinity, was punctuated by debates and controversies as alternative (and often incompatible) solutions were proposed by different mathematicians (Borasi, 1985). Even the logical­deductive method for deriving results, perceived by many as the most "solid" feature of mathematics, has encountered a number of criticisms throughout the centuries, some of which have remained unresolved (Kline, 1980).
We cannot hope that uncertainties and controversies are only a thing of the past or that future mathematicians will eventually be able to resolve all of them. Rather, mathematicians have had to accept the existence of some unavoidable limitations within the structure of mathematics itself. Starting with the creation of the first non­Euclidean geometries, mathematicians had to abandon their confidence in the absolute "truth" of even the most rigorously developed branch of their discipline and recognize, instead, that mathematics can house logically sound, yet conflicting, axiomatic systems. Even the ultimate belief that one would someday be able to verify the internal coherence and consistency of mathematics itself had to be relinquished after Godel's proof that any formal system of a certain complexity contains some undecidable propositions (Kline, 1980; Hofstadter, 1980). Most important, Lakatos's interpretation of the construction of mathematical knowledge as an iterative process of "proofs and refutations" that produces increasingly refined results (Lakatos, 1976) makes us doubt the finality of any of the mathematical results we are currently working with.
Far from affecting only the work of a small elite of professional mathematicians and philosophers, uncertainty and ambiguity pervade even the most elementary areas of mathematics, although we often do not perceive them because of the way mathematics is presented in schools and the traditional expectations about the nature of this discipline. The inquiries into tessellations and areas featured in our professional development videos provide ample evidence. Think, for example, of the role played by tentative conjectures in both experiences and of the debates that surrounded the definitions of both "tessellation" and "diamond."
As a result of all these considerations, some mathematicians have concluded that mathematical knowledge is neither absolutely true nor fully verifiable but, just as in any other science, only falsifiable and open to continuous revision:
History supports the view that there is no fixed, objective, unique body of mathematics. Moreover, if history is any guide, there will be new additions to mathematics that will call for new foundations. In this respect, mathematics is like any one of the physical sciences. Theories must be modified as new observations or new experimental results conflict with previous established theories and compel formulation of new ones. No timeless account of mathematical truth is possible. (Kline, 1980, p. 320)
Once we realize that mathematical results are neither predetermined nor absolute, we also have to accept the fact that mathematics as we know it now is as fallible as any other product of human activity. Both mathematical results and their truth are socially constructed-they are sanctioned by a community of practice (the mathematical community of the time) on the basis of agreed on criteria, which may change over time and in different contexts (Lerman, 1989). Thus, mathematical results and procedures are not totally objective. They can be influenced by cultural values, political agendas, or even just the desire to solve specific problems deemed important by the contemporary mathematics community. Alternatives to the axioms and rules that characterize accepted mathematical systems can always be devised, and the decision to accept them as part of mathematics needs to be evaluated on the basis of a number of criteria. Various non­Euclidean geometries, for example, have now been accepted as a legitimate part of mathematics along with Euclidean geometry, partly because mathematicians and physicists have realized that each geometry can be used to represent alternative models of physical space and thus can help solve problems within each context.
The following excerpts highlight some of the fundamental challenges to the common view of mathematics as the "discipline of certainty" and the ultimate example of a well­structured domain:
The history of mathematics is not one of the gradual revelation of absolute truths, but, as with all knowledge, the consequence of people's ideas, interest, conflicts and patronage, and is culturally and temporally relative. Mathematical knowledge is a social construction, the meaning of a concept such as 'polyhedron' for example, following Lakatos, is negotiated and adapted according to convention and agreement, through proofs as explanations, leading to basic refutable statements. (Lerman, 1990a, p. 27)
I have used the term humanistic to try to convey the complexity of this view of mathematics-that is, mathematics as a fallible, socially constructed, contextualized, and culture­dependent discipline driven by the human desire to reduce uncertainty but without the expectation of ever totally eliminating it. Other terms have been used to express essentially the same idea: mathematics as an ill­structured domain, mathematics as social construct, mathematics as a contextualized discipline. There is also a constructivist view of mathematics (referring to the philosophical sense of the word; see Lerman, 1989). My choice of the term humanistic has been motivated by the current usage of this term within the mathematics and mathematics education community and by my belief that emphasizing its human and humane elements can help us realize that mathematics is closer to other fields, and thus more approachable, than is usually perceived.
For many of those holding a logical positivist view of mathematics and science, the pervasiveness of this "loss of certainty" in mathematics, as Kline (1980) has characterized it, may appear quite disappointing and even somewhat disturbing. On the contrary, I would suggest that a humanistic view of mathematics not only comes closer to describing the real nature of mathematics and mathematical activity, it may also have some important benefits for mathematics students.
The recognition of limitations and "human" elements in the discipline could make it more attractive to those who have been intimidated by the absolute and authoritarian image of mathematics currently presented in schools, such as women and some other minorities. If we agree that mathematics as a discipline is not totally objective and predetermined but is influenced by economic, cultural, and even political agendas, just like any other human domain, we should question the choices made thus far about how mathematics should be covered in the precollege curriculum. We may, for example, start to look critically at the kinds of situations used to create word problems in most textbooks and see a need for alternatives:
Look at the kinds of examples we draw on in the teaching of mathematics at the moment: percentage increases in pay; simple and compound interest; hire purchase; exchange rates and angle of missile projection to hit a target. Why shouldn't we use examples to reveal prejudice and injustice, and raise children's awareness of social issues? (Lerman, 1990b, p. 29)
Once we accept the idea that mathematics is a social construct, the philosophical view of knowledge proposed by John Dewey (1993) and C. S. Peirce (cfr. Siegel & Carey, 1989) can help us appreciate the uncertainty that permeates the discipline as a positive element rather than a limitation. These philosophers suggest that uncertainty and doubt can become the motivation for inquiry and the production of knowledge. Thus, the presence of ambiguity and limitations in mathematics and in its applications should be considered as a major force for inquiry and learning, to be sought after and highlighted rather than avoided!
References
Bishop, A. (1988). Mathematical enculturation. Dordrecht, The Netherlands: Kluwer Academic Publishers.
Borasi, R. (1985). Errors in the enumeration of infinite sets. FOCUS: On learning problems in mathematics, 7 (3-4): 77-90.
Dewey, J. (1933). How we think. Boston: D. C. Heath.
Halmos, P.R. (1981). Applied mathematics is bad mathematics. In: L.A. Steen (Ed.), Mathematics tomorrow (pp. 9-20). New York: Springer-Verlag.
Hofstadter, D.R. (1980). Godel, Escher and Bach: An eternal golden braid. New York: Vintage Books.
Kline, M. (1980). Mathematics: The loss of certainty. NY: Oxford University Press.
Kline, M. (1985). Mathematics and the search for knowledge. NY: Oxford University Press.
Kuhn, T. (1970). The structure of scientific revolution. Chicago: The University of Chicago Press.
Lakatos, I. (1976). Proofs and refutations. Cambridge: Cambridge University Press.
Lave, J. (1988). Cognition in practice. Cambridge: Cambridge University Press.
Lave, J., Murtaugh, M., & de la Roche, O. (1984). The dialectic of arithmetic in grocery shopping. In: B. Rogoff & J. Lave (Eds.), Everyday cognition: Its development in social contexts. Cambridge, MA: Harvard University Press.
Lerman, S. (1989). Constructivism, mathematics and mathematics education. Educational Studies in Mathematics, 20 (2): 211-223.
Lerman, S. (1990a). A social view of mathematics: Implications for mathematics education. Humanistic Mathematics Network Newsletter, 5, 26-28.
Lerman, S. (1990b). What has mathematics to do with values ? Humanistic Mathematics Network Newsletter, 5, 29-31.
Pollak, H.O. (1970). Applications of mathematics. In: E.B.Begle (Ed.), Mathematics education (pp. 311-334). Chicago, IL: National Society for the Study of Education.
Poston, T. (1981). Purity in applications. In: L.A. Steen (Ed.), Mathematics tomorrow (pp. 9-20). New York: Springer-Verlag.
Resnick, L. (1988). Treating mathematics as an ill-structured discipline. In R. Charles & E. Silver (Eds.), The teaching and assessing of mathematical problem solving (pp. 32-60). Reston, VA: National Council of Teachers of Mathematics.
Rogoff, B. & Lave, J. (Eds.). (1984). Everyday cognition: Its development in social contexts. Cambridge, MA: Harvard University Press.
Siegel, M. & Carey, R.F. (1989). Critical thinking: A semiotic perspective. Bloomington, IN: ERIC Clearinghouse on Reading and Communication Skills.
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